1. TEST FOR RANDOMNESS: THE RUNS TEST

2. OBJECTIVE In this lecture, you will learn the following items:
How to use a runs test to analyze a series of events for randomness.

3. INTRODUCTION
Every investor wishes he or she could predict the behavior of a stock’s performance. Is there a pattern to a stock’s gain/loss cycle or are the events random? One could make a defensible argument to that question with an analysis of randomness.
The runs test (sometimes called a Wald–Wolfowitz runs test) is a statistical procedure for examining a series of events for randomness. This nonparametric test has no parametric equivalent. In this lecture, we will describe how to perform and interpret a runs test for both small samples and large samples.

4. THE RUNS TEST FOR RANDOMNESS
The runs test seeks to determine if a series of events occur randomly or are merely due to chance. To understand a run, consider a sequence represented by two symbols, A and B. One simple example might be several tosses of a coin where A = heads and B = tails.
Another example might be whether an animal chooses to eat first or drink first. Use A = eat and B = drink.

5. The first steps are to list the events in sequential order and count the number of runs. A run is a sequence of the same event written one or more times. For example, compare two event sequences. The first sequence is written AAAAAABBBBBB.
Then, separate the sequence into same groups as shown in Figure 1. There are two runs in this example, R = 2. This is a trend pattern in which events are clustered and it does not represent random behavior.

6. Consider a second event sequence written ABABABABABAB
Consider a second event sequence written ABABABABABAB. Again, separate the events into same groups (see Fig. 2) to determine the number of runs. There are 12 runs in this example, R = 12.
This is a cyclical pattern and does not represent random behavior either. As illustrated in the two examples earlier, too few or too many runs lack randomness.

7. A run can also describe how a sequence of events occurs in relation to a custom value. Use two symbols, such as A and B, to define whether an event exceeds or
falls below the custom value. A simple example may reference the freezing point of water where A = temperatures above 0°C and B = temperatures below 0°C.
In this example, simply list the events in order and determine the number of runs as described earlier.

8. After the number of runs is determined, it must be examined for significance.
We may use a table of critical values (see Table B.10). However, if the numbers of values in each sample, n1 or n2, exceed those available from the table, then a large sample approximation may be performed.

9. Example 1
Runs Test (Small Data Samples)
The following study seeks to examine gender bias in science instruction. A male science teacher was observed during a typical class discussion.
The observer noted the gender of the student that the teacher called on to answer a question. In the course of 15 min, the teacher called on 10 males and 10 females. The observer noticed that the science teacher called on equal numbers of males and females, but he wanted to examine the data for a pattern.

10. To determine if the teacher used a random order to call on students with regard to gender, he used a runs test for randomness.
Using M for male and F for female, the sequence of student recognition by the teacher is:
MFFMFMFMFFMFFFMMFMMM.

11. 1. State the Null and Research Hypotheses
The null hypothesis states that the sequence of events is random. The research hypothesis states that the sequence of events is not random.
The null hypothesis is
H0: The sequence in which the teacher calls on males and females is random.
The research hypothesis is
HA: The sequence in which the teacher calls on males and females is not random.

13. 3. Choose the Appropriate Test Statistic
The observer is examining the data for randomness. Therefore, he is using a runs test for randomness.

14. Compute the Test Statistic
First, determine the number of runs, R. It is helpful to separate the events as shown in Figure 3. The number of runs in the sequence is R = 13.

15. 5. Determine the Value Needed for Rejection of the Null Hypothesis Using the Appropriate Table of Critical Values for the Particular Statistic
Since the sample sizes are small, we refer to Table B.10, which lists the critical values for the runs test. There were 10 males (n1) and 10 females (n2). The critical values are found on the table at the point for n1 = 10 and n2 = 10.
We set = The critical region for the runs test is 6 < R < 16. If the number of runs, R, is 6 or less, or 16 or greater, we reject our null hypothesis.

16. 6. Compare the Obtained Value with the Critical Value
We found that R = 13. This value is within our critical region (6 < R < 16).
Therefore, we do not reject the null hypothesis.

17. 7. Interpret the Results
We did not reject the null hypothesis, suggesting that the sequence of events is random. Therefore, we can state that the order in which the science teacher calls on males and females is random.

18. 8 Reporting the Results
The reporting of results for the runs test should include such information as the sample sizes for each group, the number of runs, and the p-value with respect to .
For this example, the runs test indicated that the sequence was random (R = 13, n1 = 10, n2 = 10, p > 0.05). Therefore, the study provides evidence that the science teacher was demonstrating no gender bias.

19. Example 2
Runs Test Referencing a Custom Value
The science teacher in the earlier example wishes to examine the pattern of an “at-risk” student’s weekly quiz performance. A passing quiz score is 70. Sometimes the student failed and other times he passed. The teacher wished to determine if the student’s performance is random or not. Table 1 shows the student’s weekly quiz scores for a 12-week period.

26. SUMMARY
The runs test is a statistical procedure for examining a series of events for randomness. This nonparametric test has no parametric equivalent. In this lecture, we described how to perform and interpret a runs test for both small samples and large samples.